Using the electrons spectra calculated in the previous Section, it is possible to calculate the non-thermal SZE, using the full relativistic formalism [1]}, [2]}, where the intensity of the SZE is given by
\(\Delta I (x) = \tau [J_1(x)-I_0(x)]\) .
Here \(x=h\nu /(k_B T_{0})\)  is the frequency normalized at the CMB temperature \(T_0\) , \(I_0(x)\)  is the unperturbed CMB spectrum, \(\tau \)  is the electrons optical depth, proportional to the electrons density integrated along the line of sight, and \(J_1(x)\)  is calculated by convolving the CMB spectrum with the single scattering redistribution function, which is given by the convolution of the electrons spectrum with the Compton scattering redistribution function for an electron with a given energy [3]}.
